President Obama and his aides found themselves scrambling Monday to assure skeptics and defend the deal struck over the weekend to curb Iran’s nuclear program while providing the Islamic republic with some $5.7 billion in relief from economic sanctions.

In the face of rising skepticism from top lawmakers of both parties and outright opposition from Israel, Mr. Obama touted the deal’s upside in remarks before a speech on immigration in San Francisco, contending that, “for the first time in a decade, we’ve halted the progress on Iran’s nuclear program.”

Acknowledging there were many diplomatic hurdles still to clear, Mr. Obama told the crowd, “we cannot close the door on diplomacy, and we cannot rule out peaceful solutions to the world’s problems … Tough talk, and bluster may be the easy thing to do politically, but it is not the right thing for our security.”

Behind the scenes, insiders in the administration were quick to go a step further, claiming that despite the high-dollar figure agreed to by U.S. negotiators in Geneva, the Iranian economy will barely feel any relief.

“The vast architecture of our sanctions will remain in place and we will continue to vigorously enforce it,” said one official inside the Treasury Department, where the powerful and discreet Office of Foreign Assets Control has spent much of the past four years erecting the Obama administration’s regime of sanctions against Iran and engineering a global embargo on the republic’s oil industry.

The Treasury official, who agreed to issue a statement to The Washington Times only on condition of anonymity, asserted that $4.2 billion in oil sales and $1.5 billion in other export revenues that Secretary of State John F. Kerry has said Iran will have access to under the Geneva deal is not going to “provide meaningful impact in Iran.”

“They will actually be under much more economic pressure at the end of the six months than they are today,” said the official, whose remarks came as criticism of deal mounted among several key lawmakers in Washington from both sides of the aisle.

At issue, the lawmakers say, is the fact Iran is being permitted under the deal to carry on with some degree — albeit significantly truncated — of uranium enrichment, the core activity in any potential nuclear bomb development program.

While Iran has agreed to dilute or convert its entire existing stockpile of 20 percent enriched uranium, the agreement lets Iranian scientists proceed with current enrichment activities to a level of 5 percent.

Iranian leaders have long claimed the enrichment activities are for such peaceful purposes as generating civilian electricity. But the U.S. and its allies, particularly Israel, say Iran may be close to developing a bomb.

After harshly criticizing the deal reached in Geneva, Israeli Prime Minister Benjamin Netanyahu said Monday that he had spoken by telephone with President Obama, and that an envoy led by Israel’s National Security Adviser Yossi Cohen would be traveling to Washington during the coming days to hold talks about the situation with administration officials.

Apart from the enrichment issue, one influential Republican lawmaker suggested Monday that the Obama administration may be actively misleading the public about the true scope of the sanctions relief that Iran will experience during the months ahead.

Rep. Mike Rogers, Michigan Republican and chairman of the House Intelligence Committee, told CNN that the sanctions relief may actually be four times as large as Mr. Kerry claimed when he announced the deal over the weekend.

Instead of $5.7 billion, Mr. Rogers said analysts think the deal could be worth as much as $20 billion by the time it is fully implemented because it allows Iran to import previously banned gold and other precious metals, which can facilitate previously blocked international trade.

“The reason this is a big deal is because this is the way they can barter for other things, and they’ve done this with other nations before,” he said. “They’ve been trying to get around sanctions using the barter system, and they’ve done it with other ways, having, you know, more money on one side and one side trading a different account.”

Iranian authorities, said Mr. Rogers, have already used trade in gold to explore “all different ways to get around” the existing sanctions on other areas of the Iranian economy, such as its oil industry. The Obama administration’s lack of forthrightness about this “worries me,” said Mr. Rogers.